MULTI-BUMP GROUND STATES OF THE FRACTIONAL
GIERER-MEINHARDT SYSTEM ON THE REAL LINE
JUNCHENG WEI AND WEN YANG
Abstract. In this paper we study ground-states of the fractional GiererMeinhardt system on the line, namely the solutions of the problem

2

 (−∆)s u + u − uv = 0,
in R,
s
2s
(−∆) v + ε v − u2 = 0,
in R,

 u, v > 0, u, v → 0
as |x| → +∞.
We prove that given any positive integer k, there exists a solution to this problem for s ∈ [ 21 , 1) exhibiting exactly k bumps in its u−component, separated
1−2s
1
from each other at a distance O(ε 4s ) for s ∈ ( 12 , 1) and O(| log ε| 2 ) for
s = 12 respectively, whenever ε is sufficiently small. These bumps resemble the
shape of the unique solution of
(−∆)s U + U − U 2 = 0,
0 < U (y) → 0 as |y| → ∞.
1. Introduction
In this paper we consider the following fractional Gierer-Meinhardt system in R
{
2
(−∆)s u + u − uv = 0
in R,
(1.1)
(−∆)s v + ε2s v − u2 = 0 in R,
where (−∆)s , 0 < s < 1, denotes the fractional Laplace operator. (For the definition, See Section 2 below.)
When s = 1, this is the classical Gierer-Meinhardt system proposed by GiererMeinhardt in [12] in 1972. More precisely they considered the following reactiondiffusion system as a model of biological pattern formation

2
 at = d∆a − a + ah ,
in Ω × (0, t),
(1.2)
ht = D∆h − h + a2 , in Ω × (0, t),

∂ν a = ∂ν h = 0,
on ∂Ω × (0, t),
where d, D > 0 are diffusion rates, Ω ⊂ Rn is a bounded domain and ∂ν denotes
the derivative in the outer normal direction. The Gierer-Meinhardt system was
used in [12] to model head formation of Hydra, an animal of a few millimeters in
length, made up of approximately 100,000 cells of about fifteen different types. It
consists of a ”head” region located at one end along its length. Typical experiments with hydra involve removing part of the ”head” region transplanted area is
sufficiently far from the (old) head. These observations led to the assumption of
the existence of two chemical substances a slowly diffusing activator and a rapidly
diffusing inhibitor, whose concentrations at the point x ∈ Ω and time t > 0 are
represented, respectively, by the quantities a(x, t) and h(x, t). Their diffusion rates,
given by the positive constants d and D are then assumed to be that d ≪ D. The
Gierer-Meinhardt system falls within the framework of a theory proposed by Turing
1
2
JUNCHENG WEI AND WEN YANG
[27] in 1952 as a mathematical model for the development of complex organisms
from a single cell. He speculated that localized peaks in concentration of chemical
substances, known as inducers or morphogens, could be responsible for a group of
cells developing differently from the surrounding cells. Turing discovered through
linear analysis that a large difference in relative size of diffusivities for activating
and inhibiting substances carries instability of the homogeneous, constant steady
state, thus leading to the presence of nontrivial, possibly stable stationary configurations. Activator-inhibitor systems have been used widely in the mathematical
theory of biological pattern formation [17, 18]. Substantial research concerning this
system has been generated in recent years. We refer the reader to the survey papers
[24], [35] and the book [36] for the overview of the subject.
In the last twenty years there have been intensive research on the existence and
stability of steady states of the Gierer-Meinhardt system (1.2) in a bounded domain.
It is known that there are multiple spikes solutions which may be stable. We refer
to papers [6], [15], [30], [28], [33], [34] and the book [36] and the references therein.
In the case of the domain being the whole space, after suitable rescaling the steady
state problem of (1.2) becomes

2
 ∆u − u + uv = 0, u > 0, in Rn
(1.3)
∆v − ε2 v + u2 = 0, v > 0, in Rn

u, v → 0 as |x| → +∞.
A solution to (1.3) is called ground state. In the real line case the existence of single
and multiple pulse solutions is proved independently by Doelman-Gardner-Kaper
in [7] (via geometric dynamical system method) and by Chen-del Pino-Kowalczyk
in [5] (via PDE reduction method). Similar results have been obtained for the case
n = 2 by del Pino-Kowalczyk-Wei in [8]. In higher dimensional case another type
of solutions exist: solutions which are radially symmetric but have rings concentrations. We refer to Ni-Wei [25], Kolokolnikov-Wei [21] and Kolokolnikov-Wei-Yang
[20]. In R3 there exists also axially symmetric solution with smoke ring concentrations. See Kolokolnikov-Ren [22]. The presence of steady configurations in the
d
whole space appears driven by smallness of the relative size ε2 = D
of the diffusion
rates of the activating and inhibiting substances. The present paper deals with the
case of equation (1.3) in the nonlocal diffusion–fractional laplacian case (1.1) for
s ∈ [ 12 , 1). Next we briefly discuss the fractional laplacian and nonlocal diffusion.
In probability, we consider the random walk for Lévy processes:
∑
un+1
=
Pjk unk ,
j
k
where Pjk denotes the transition function which has a tail (i.e., power decay with the
distance |j − k|). By taking the limit we get an operator (−∆)s as the infinitesimal
generator of a Lévy process: if Xt is the isotropic 2s−stable lévy process we have
1
(−∆)s u(x) = lim+ E[u(x) − u(x + Xh )].
h→0 h
When s = 1 it corresponds to the Brownian motion.
Fractional diffusion equations have been used to model anomalously slow or fast
scattering of particles in a variety of natural applications. A consideration of the
problem of anomalous sub-diffusion with reactions in terms of continuous-time random walks (CTRWs) with sources and sinks leads to a fractional activator-inhibitor
FRACTIONAL GIERER-MEINHARDT SYSTEM
3
model with a fractional order temporal derivative operating on the spatial Laplacian. A similar type of system has also been proposed for diffusion with reactions
on a fractal. The problem of anomalous super-diffusion with reactions has also been
considered and in this case a fractional reaction-diffusion model has been proposed
with the spatial Laplacian replaced by a spatial fractional differential operator. If
the reaction time is not short compared with the diffusion time in sub-diffusive
systems with reactions (for example, if many encounters between reactants are required before reactions proceed), then an alternate model has been proposed where
the fractional order temporal derivative operates on both the spatial Laplacian and
the reaction term. An important distinction between the two anomalous reactiondiffusion models becomes apparent when the concentration of species is spatially
homogeneous, this latter model does not reduce to the classical macroscopic rate
equations except when the diffusion is also non-anomalous. For the more background on fractional reaction-diffusion system, we refer the readers to [13, 14, 16]
and references therein. We shall mention that weakly nonlinear analysis has been
done to constant equilibriums for Turing’s system by Henry-Langlands-Wearne [14]
and Golovin-Matkowsky-Volpert [13]. In [23] the author studied a slightly different
Gierer-Meinhardt system with fractional diffusion

2
 at = −d(−∆)s a − a + ah , in (−1, 1) × (0, t),
2
(1.4)
h = D∆h − h + a ,
in (−1, 1) × (0, t),
 t
ax (±1, t) = hx (±1, t) = 0
where (−∆)s denotes the spectral fractional Laplacian.
In this paper we shall consider the existence of nonlinear patterns for the classical
Gierer-Meinhardt system in the real line (1.1) with fractional diffusion. A similar
notable feature of problem (1.1) (as the classical case) is, it will be shown in this
work, the presence of a large number of solutions (modulo translations) as the
parameter ε gets smaller. More precisely, given any positive integer k we find a
number εk such that if 0 < ε < εk then there exists a solution exhibiting exactly
k bumps in the activator. Besides, after appropriate re-scaling of u, these bumps
1−2s
are approaching a universal profile and are separately from each other ε 4s and
1
(− log ε) 2 for the case s ∈ ( 12 , 1) and s = 12 respectively. We remark that this
phenomenon is intrinsic to the full system, for only one ground state of the following
equation modulo translations exists ([9])
(−∆)s U + U − U 2 = 0 in R,
0 < U (y) → 0 as |y| → ∞.
(1.5)
Before stating our main results on the existence of such solutions, we need the following preparations. In the sequel by U (x) we denote the unique radially symmetric
solution of (1.5). (For the existence and uniqueness, we refer to Frank-Lenzmann
[9].) In order to describe the position of bumps, we set
m (
)
∑
αs U (2qi ) + βs ε2s−1 |2qi |2s−1
Ξs (q1 , q2 , · · · , qm ) =
i=1
+
∑(
αs U (qi − qj ) + βs ε2s−1 |qi − qj |2s−1
i̸=j
+
∑(
i̸=j
αs U (qi + qj ) + βs ε2s−1 |qi + qj |2s−1
)
)
(1.6)
4
JUNCHENG WEI AND WEN YANG
for s ∈ ( 12 , 1) and
Ξ 12 (q1 , q2 , · · · , qm ) =
m (
∑
α 21 U (2qi ) + β 21
i=1
∑(
)
1
1 log |2qi |
log ε
)
1
log
|q
−
q
|
i
j
log 1ε
i̸=j
)
∑(
1
+
α 12 U (qi + qj ) + β 12
log
|q
+
q
|
i
j
log 1ε
i̸=j
+
α 12 U (qi − qj ) + β 12
(1.7)
for s = 12 .
In Section 8, we shall prove the function Ξs , s ∈ [ 12 , 1) admits global minimal
point in the interior of the following set
{
}
1−2s
1−2s
1 1−2s
Qs,η = (q1 , q2 , · · · , qm ) | ε 4s > qi > ηε 4s , |qi − qj | > ηε 4s
(1.8)
η
for s ∈ ( 12 , 1) and
{
1 1
1
1 1
1 1}
Q 12 ,η = (q1 , q2 , · · · , qm ) | (log ) 2 > qi > η(log ) 2 , |qi − qj | > η(log ) 2
η
ε
ε
ε
(1.9)
for s = 12 respectively. Here η denotes a small positive number and with the
constants αs , βs being given later.
We denote one of the global minimal points (if there are more) of Ξs in Qs,η by
qs = (qs,1 , qs,2 , · · · , qs,m ).
Remark: Since the function Ξs (q1 , q2 , · · · , qm ) is analytic in Qs,η , therefore, all
the global minimal points are discrete.
Let us set
 (
)−1
∫

k
1−2s

U 2 (y)dy
, s ∈ ( 21 , 1),
π ε
2s sin( 2s
)
R1
)−1
(
τε =
∫

 πk log 1ε 1 U 2 (y)dy
,
s = 12 .
R
(1.10)
Our main result is the following.
Theorem 1.1. Let k = 2m, m ≥ 1 be a fixed positive integer. There exists εk > 0
such that for each 0 < ε < εk , problem (1.1) admits a solution (u, v) with the
following property:
m
∑
(
)
−1
lim τε uε (x) −
U (x − qs,i ) + U (x + qs,i ) = 0,
ε→0
i=1
uniformly in x ∈ R , while for the second component v, we have
)
(
lim τε−1 vε (qs,i + x) − 1 + τε−1 vε (qs,i − x) − 1 = 0
1
ε→0
for any i uniformly in compact sets in x, i = 1, 2, · · · , m. A similar result holds
when k = 2m + 1, m ≥ 0.
FRACTIONAL GIERER-MEINHARDT SYSTEM
5
We will only give the details of constructing solutions exhibiting an even number
of bumps in the line. The case of odd number of bumps can be treated in a similar
manner, and we will discuss the necessary changes in section 8.
The method employed in the proof of Theorem 1.1 consists of a LyapunovSchmidt type reduction. Fixing m points which from Qs,η with η being determined
later, an auxiliary problem is solved uniquely, and solutions satisfying the required
conditions will be those precisely satisfying a nonlinear system of equations of the
form
cs,i (q1 , q2 , · · · , qm ) = 0, i = 1, 2, · · · , m,
where for such a class of points the function cs,i satisfy
∑(
)]
∂ [1
cs,i (q1 , q2 , · · · , qm ) =
Fs (|2qi |) +
Fs (|qi − qj |) + Fs (|qi + qj |) + σs,i ,
∂qi 2
j̸=i
(1.11)
where
{
Fs (r) =
αs U (r) + βs ε2s−1 r2s−1 ,
log r
α 12 U (r) + β 21 log
1 ,
ε
and
{
σs,i =
o(1)εs+ 2 − 2s ,
3
o(1)(− log ε)− 2 ,
1
1
s ∈ ( 21 , 1),
s = 12
s ∈ ( 21 , 1),
s = 12 .
We can easily find that solutions of the problem cs,i = 0 are closely related to
critical points of the functional defined in (1.6) and (1.7) for s ∈ ( 12 , 1) and s = 12
respectively. In the classical case, since the profile U of each bump is exponential
decay, in the leading term of ci , we only need to consider the neighbor points of qi .
However, in the fractional case, the profile U has algebraic decay. As a result, all the
points interact with each other strongly and we have to consider all the interactions
in dealing with cs,i . Even though the reduced problem is a finite dimensional one,
we can not show that the locations of the bumps are unique and nondegenerate.
In the classical case s = 1, the locations can be computed explicitly and the nondegeneracy can be proved easily. This is the main difficulty and new feature in
considering (1.1).
It is clear that the results of Theorem 1.1 can be extended without any difficulty
to Gierer-Meinhardt system in which activator and inhibitor have different diffusion
characters
{
2
(−∆)s1 u + u − uv = 0
in R,
(1.12)
s2
2s2
(−∆) v + ε v − u2 = 0 in R,
where s1 ∈ (0, 1] and s2 ∈ [ 21 , 1]. It is also possible to generalize to Gierer-Meinhardt
system with more general nonlinearity. We omit the details.
The rest of the paper will be devoted to the proof of Theorem 1.1. In section 2 we
study the fractional Laplacian operator and the behavior of the Green function for
(−∆)s + I. In section 3 we set up the scheme of proof, in particular we explain why
the constant τε is the right scaling factor to get the desired multi-bump expansion.
The program outlined there is carried over the following sections.
6
JUNCHENG WEI AND WEN YANG
2. Preliminaries
In this section, we provide some elementary properties of the operators (−∆)s +I.
Let 0 < s < 1. Various definitions of the fractional Laplacian (−∆)s ϕ of a function
ϕ defined in Rn are available, depending on its regularity and growth properties.
For ϕ ∈ H 2s (Rn ), the standard definition is given via Fourier transform b .
(−∆)s ϕ ∈ L2 (Rn ) is defined by the formula
\s ϕ.
|ξ|2s ϕ̂(ξ) = (−∆)
(2.1)
When ϕ is assumed in addition sufficiently regular, we obtain the direct representation
∫
ϕ(x) − ϕ(y)
(−∆)s ϕ(x) = ds,n
dy
(2.2)
n+2s
n
R |x − y|
for a suitable constant ds,n and the integral is understood in a principal value sense.
This integral makes sense directly when s < 21 and ϕ ∈ C 0,α (Rn ) with α > 2s, or if
ϕ ∈ C 1,α (Rn ), 1 + α > 2s. In the latter case, we can desingularize the integral and
represent it in the form
∫
ϕ(x) − ϕ(y) − ∇ϕ(x)(x − y)
(−∆)s ϕ(x) = ds,n
dy.
|x − y|n+2s
Rn
Another useful (local) representation, found by Caffarelli and Silverstre [2], is via
the following boundary value problem in the half space Rn+1
= {(x, y) | x ∈ Rn , y >
+
0} :
{
∇ · (y 1−2s ∇ϕ̃) = 0 in Rn+1
+ ,
ϕ̃(x, 0) = ϕ(x)
on Rn .
Here ϕ̃ is the s−harmonic extension of ϕ, explicitly given as a convolution integral
with the s−Poisson kernel ps (x, y),
∫
ϕ̃(x, y) =
ps (x − z, y)ϕ(z)dz,
Rn
where
ps (x, y) = Cn,s
y 4s−1
(|x|2 + |y|2 )
n−1+4s
2
∫
and Cn,s achieves Rn p(x, y) = 1. Then under suitable regularity, (−∆)s ϕ is the
Dirichlet-to-Neumann map for this problem, namely
(−∆)s ϕ(x) = lim+ y 1−2s ∂y ϕ̃(x, y).
(2.3)
y→0
Characterizations (2.1)-(2.3) are all equivalent for instance in Schwartz’s space of
rapidly decreasing smooth functions.
Now let us consider for a number m > 0 and g ∈ L2 (Rn ) the equation
(−∆)s ϕ + mϕ = g in Rn .
Then in terms of Fourier transform, this problem, for ϕ ∈ L2 , reads
(|ξ|2s + m)ϕ̂ = ĝ
and has a unique solution ϕ ∈ H 2s (Rn ) given by the convolution
∫
ϕ(x) = Tm [g] :=
G(x − z)g(z)dz,
Rn
(2.4)
FRACTIONAL GIERER-MEINHARDT SYSTEM
7
where
1
.
|ξ|2s + m
Using the characterization (2.3) written in weak form, ϕ can be characterized by
ϕ(x) = ϕ̃(x, 0) in trace sense, where ϕ̃ ∈ H is the unique solution of
∫ ∫
∫
∫
ϕφ =
gφ, for all φ ∈ H,
(2.5)
∇ϕ̃∇φy 1−2s + m
b
G(ξ)
=
Rn+1
+
Rn
Rn
1
where H is the Hilbert space of functions φ ∈ Hloc
(Rn+1
+ ) such that
∫ ∫
∫
∥φ∥2H :=
|φ|2 < +∞,
|∇φ|2 y 1−2s + m
Rn+1
+
Rn
or equivalently the closure of the set of all functions in Cc∞ (Rn+1
+ ) under this norm.
A useful fact for our purpose is the equivalence of the representations (2.4) and
(2.5) for g ∈ L2 (Rn ).
Lemma 2.1. Let g ∈ L2 (Rn ). Then the unique solution ϕ̃ ∈ H of problem (2.5) is
given by the s−harmonic extension of the function ϕ = Tm [g] = G ∗ g.
For a proof, one can see Lemma 2.1 in [4]. Let us recall the main properties
of the fundamental solution G(x) in the representation (2.4), which are stated for
instance in [11] and [10].
We have that G is radially symmetric and positive, G ∈ C ∞ (Rn \ {0}) satisfying
•
C
|G(x)| + |x||∇G(x)| ≤
for all |x| ≤ 1,
|x|n−2s
•
lim G(x)|x|n+2s = γ > 0,
|x|→∞
•
|x||∇G(x)| ≤
C
|x|n+2s
for all |x| ≥ 1.
In next lemma, we get a more specific behavior of the Green function G(x)
around x = 0 for n = 1.
Lemma 2.2. Let G(x) be the Green function of the following equation
(−∆)s G(x) + G(x) = δ(x).
Then, we have
{
a0 + a1 |x|2s−1 + O(|x|min{2,4s−1} ) as |x| → 0,
G(x) =
− π1 log |x| + a2 + O(|x|) as |x| → 0,
(2.6)
if s ∈ ( 12 , 1),
if s = 12 ,
(2.7)
where a0 , a1 , a2 will be given in the proof.
Remark: a1 is negative.
Proof. By using fourier transform, we can write the equation (2.6) as
b
(|ξ|2s + 1)G(ξ)
= 1.
Therefore, we have
G(x) =
1
2π
∫
∞
−∞
eixξ
1
dξ =
1 + |ξ|2s
π
∫
0
(2.8)
∞
cos(xξ)
dξ.
1 + |ξ|2s
(2.9)
8
JUNCHENG WEI AND WEN YANG
We note G(x) is an even function, in the following, we only need to consider the
behavior of G(x) when x → 0+ . We divide our discussion into two parts.
If s ∈ ( 12 , 1), we have
1
π
∫
∫ ∞
)
2
(sin xξ
1
2 )
dξ
−
2
dξ
1 + |ξ|2s
1 + |ξ|2s
0
0
∫ ∞
xξ 2
(sin 2 )
1
2
dξ
π −
2s sin( 2s ) π 0 1 + |ξ|2s
∫
2 2s−1 ∞ (sin 2t )2
1
x
dt
π −
) π
2s sin( 2s
t2s + x2s
0
∫
x2s (sin 2t )2 )
1
2 2s−1 ∞ ( (sin 2t )2
−
x
−
dt
π
2s sin( 2s
) π
t2s
t2s (t2s + x2s )
0
∫
(sin 2t )2
2 4s−1 ∞
2s−1
a0 + a1 x
+ x
dt.
(2.10)
π
t2s (t2s + x2s )
0
cos(xξ)
1(
dξ
=
1 + |ξ|2s
π
∞
0
=
=
=
=
∫
∞
where
a0 =
1
2
π , a1 = −
2s sin( 2s
)
π
∫
∞
0
(sin 2t )2
2
dt = − sΓ(−2s) sin(πs) < 0.
t2s
π
We can write the last term on the right hand side of (2.10) as
∫
(
x4s−1 (sin 2t )2
dt
=
t2s (t2s + x2s )
∞
0
∫
∫
x
+
0
∫
1
∞
+
x
1
) x4s−1 (sin t )2
2
dt.
t2s (t2s + x2s )
For each term on the right hand side of (2.11), we have
∫
x
0
∫
1
x
∫
∞
1
x4s−1 (sin 2t )2
dt ≤ Cx2s−1
t2s (t2s + x2s )
x4s−1 (sin 2t )2
dt ≤ Cx4s−1
t2s (t2s + x2s )
x4s−1 (sin 2t )2
dt ≤ Cx4s−1
t2s (t2s + x2s )
∫
1
x
∫
1
∫
0
x
t2
dt = O(x2 ),
t2s
t2
dt = O(x2 + x4s−1 ),
t4s
∞
(sin 2t )2
dt = O(x4s−1 ).
t4s
Hence, we have
∫
0
∞
x4s−1 (sin 2t )2
dt = O(xmin{2,4s−1} ),
t2s (t2s + x2s )
as a result, we get the first one in (2.7).
(2.11)
FRACTIONAL GIERER-MEINHARDT SYSTEM
9
When s = 12 , we have
π
∫ ∞
∫ 2x
∫ ∞
cos(xξ)
cos(xξ)
cos(xξ)
dξ =
dξ +
dξ
π
1
+
ξ
1
+
ξ
1+ξ
0
0
2x
π
∫ 2x
∫ ∞
( 1
2)
2(sin xξ
cos(xξ)
2 )
=
−
dξ +
dξ
π
1
+
ξ
1
+
ξ
1+ξ
0
2x
∫ π2
∫ ∞
(sin 2t )2
π
cos t
= log(1 +
)−2
dt +
dt
π
2x
x+t
x+t
0
2
∫ π2
∫ ∞
(sin 2t )2
π
cos t
= − log x + log − 2
dt +
dt
π
2
t
t
0
2
∫ π2
∫ ∞
(sin 2t )2
cos t
+ 2x
dt − x
dt + O(x).
π
t(x + t)
t(x + t)
0
2
It is easy to see,
∫
π
2
0
Therefore, for s =
1
2,
(sin 2t )2
dt,
t(x + t)
a2 =
∞
π
2
cos t
dt = O(1).
t(x + t)
we obtain
G(x) = −
where
∫
1
log x + a2 + O(x) as x → 0+ ,
π
1(
π
log − 2
π
2
∫
π
2
0
(sin 2t )2
dt +
t
∫
∞
π
2
(2.12)
cos t )
dt .
t
3. The scheme of the proof
Our strategy of the proof of the main results is based on the idea of solving the
second equation in (1.1) for v and then working with a nonlocal elliptic PDE rather
than directly with the system. It is convenient to do this by replacing first u by
τε u and v by τε v, which transforms (1.1) into the problem

2
 (−∆)s u + u − uv = 0
in R,
s
2s
(3.1)
(−∆) v + ε v − τε u2 = 0 in R,

u, v > 0, u, v → 0,
as |x| → ∞,
with the choice of the parameter τε as in (1.10),
 (
)−1
∫

k
1−2s
2

U
(y)dy
, s ∈ ( 21 , 1),
π ε
2s sin( 2s )
R1
(
)
τε =
−1
∫

 πk log 1ε 1 U 2 (y)dy
,
s = 12 .
R
Then, we have
u∼
k
∑
i=1
U (x − qi ),
v ∼ 1,
(3.2)
10
JUNCHENG WEI AND WEN YANG
i.e., the height of the bumps near the qi remains bounded as ε → 0.
In the sequel, by T (h) we denote the unique solution of the equation
(−∆)s v + ε2s v = τε h in R, v(x) → 0 as |x| → +∞,
)−1
(
. Solving the second equation for v
for h ∈ L2 (R), namely T = τε (−∆)s + ε2s
in (3.1) we get v = T (u2 ), which tends to the nonlocal equation
(−∆)s u + u −
u2
= 0.
T (u2 )
(3.3)
We consider points q1 , q2 , · · · , qk , k = 2m in R which are the candidates for the
location of spikes. These points (q1 , q2 , · · · , qk ) are restricted in the following set
{
(q1 , q2 , · · · , qk ) ∈ Λs = (q1 , q2 , · · · , qk ) | qi = −qk+1−i , q1 > q2 > · · · > qk ,
}
(q1 , q2 , · · · , qm ) ∈ Qs,η .
(3.4)
Let us write
W (x) =
k
∑
U (x − qi ).
i=1
We look for a solution to (3.3) in the form u = W + ϕ, where ϕ is a lower order
term. Then, formally, we have
T (u2 ) = T (W 2 ) + 2T (W ϕ) + h.o.t.,
where h.o.t. corresponds to the higher order terms. We denote V = T (W 2 ). By
Gε (|y|) we denote the fundamental solution to (−∆)s + ε2s in R. We can write
∫
k ∫
∑
T (W 2 ) = τε W 2 (y)Gε (|x − y|)dy ∼ τε
U 2 (x − qi )Gε (|x − y|)dy,
i=1
where the integration extends over all R. For |x| = o(ε−1 ) we have
{
α0 ε1−2s + α1 |x|2s−1 + O(ε3−2s |x|2 + ε2s |x|4s−1 ), if s ∈ ( 12 , 1),
Gε (|x|) =
− π1 log(ε|x|) + O(1),
if s = 12 .
Using this, and the definition of τε we get that near the qi ’s,
V (x) = 1 + h.o.t..
Arguing similarly we get
∫
T (W ϕ) = ω
W ϕ + h.o.t., ω =
1
∫
.
k U2
Then
u2
W 2 + 2W ϕ + h.o.t.
W2
=
=
+ 2W ϕ − 2ωW 2
v
V + 2T (W ϕ) + h.o.t.
V
∫
W ϕ + h.o.t..
Substituting all this into (3.3) we obtain the equation for ϕ
∫
s
2
W ϕ = S(W ) + N (ϕ),
(−∆) ϕ + (1 − 2W )ϕ + 2ωW
where S(W ) = −(−∆)s W − W +
W2
V
and N (ϕ) is defined by
∫
(W + ϕ)
W2
2
W ϕ,
N (ϕ) =
−
− 2W ϕ + 2ωW
T ((W + ϕ)2 )
V
2
(3.5)
FRACTIONAL GIERER-MEINHARDT SYSTEM
11
represents higher order terms in ϕ.
Thus we have reduced the problem of finding solutions to (3.1) to the problem
of solving (3.5) for ϕ. We set ∂W
∂qj = Zj . Rather than solving problem (3.5), we
consider first the following auxiliary problem: given points qj , find a function ϕ
such that for certain constants ci the following equation is satisfied
∑
Lϕ = S(W ) + N (ϕ) +
cj Zj , ⟨ϕ, Zj ⟩ = 0, j = 1, 2, · · · , k,
(3.6)
j
where
∫
Lϕ = (−∆)s ϕ + (1 − 2W )ϕ + 2ωW 2
Wϕ
and ⟨·, ·⟩ denotes the L2 inner product.
We will prove in section 4 that this problem is uniquely solvable within a class of
small functions ϕ for all points (q1 , q2 , · · · , qk ) satisfying constraints (3.4). Besides,
the resulting constants ci (q1 , q2 , · · · , qk ) admit the expansion (1.11) . We will of
course get a solution of the full problem whenever the points qi are adjusted in such
a way that all of ci ’s vanish. We show the existence of such points in section 8,
where the main result is finally established. In remainder of the paper we rigorously
carry out the program outlined above. In particular, we will need to understand
the invertibility properties of the linear operator L first. We will do it in the next
section.
4. The linear operator
Proposition 4.1. Let U be the unique, positive, radially symmetric solution to
(−∆)s U + U − U 2 = 0.
(4.1)
(a) There exists a positive constant ds depending on s only such that, as |x| →
∞, the following formula holds
U (|x|) =
bs
(1 + o(1)).
|x|1+2s
Moreover, U ′ (x) < 0 for x > 0 and
(1 + 2s)bs
(1 + o(1)) as x → +∞.
x2+2s
(b) Let L0 = (−∆)s + (1 − 2U )id. Then we have
{ ∂U }
.
Ker(L0 ) = span
∂x
(c) Let L be the operator defined in (3.6) and let
∫
L∗ = (−∆)s ϕ + (1 − 2W )ϕ + 2ωW W 2 ϕ
U ′ (x) = −
be its formal disjoint. If we denote
Zj =
∂W
,
∂qj
12
JUNCHENG WEI AND WEN YANG
then, for all j = 1, 2, · · · , k, we have
LZj = O(min |qi − qj |−(2s+1) )W (x)2 + O(min |qi − qj |−(2s+2) )W (x),
i̸=j
i̸=j
∗
−(2s+2)
L Zj = O(min |qi − qj |
i̸=j
)W (x).
Proof. The proof of part (a) and (b) is given in [9, Proposition 1.1 and Theorem 2.3].
2
In fact, it was proven in [1] that (apart from translations) the function U (x) = 1+x
2
1
1
is the only positive solution of (4.1) in H 2 (R) when s = 2 . Part (c) is a consequence
of direct computations.
We shall carry out the analysis of the linear operator L in a framework of
weighted L∞ spaces. For this purpose we consider the following norm for a function
defined in R: given points q1 , q2 , · · · , qk we define
∥ϕ∥∗ = ∥ρ(x)−1 ϕ∥L∞ (R) ,
where
ρ(x) =
k
∑
j=1
(4.2)
1
1
,
< µ ≤ 1 + 2s.
(1 + |x − qj |)µ 2
We first consider a problem that will later give rise to the finite-dimensional
reduction. Given a function h, ∥h∥∗ < ∞ find a ϕ and constants cj , j = 1, 2, · · · , k
such that one has

∑
 Lϕ = h + j cj Zj in R,
(4.3)
ϕ(x) → 0 as |x| → ∞,

⟨ϕ, Zj ⟩ = 0 for j = 1, 2, · · · , k.
By c we will denote a vector with components cj .
We refer to a pair (ϕ, c) as a solution to (4.3). We have the following existence
result for (4.3).
Theorem 4.2. There exists positive numbers R and C such that, for any points
qj satisfying |qi − qj | > R for all i ̸= j, and h with ∥h∥∗ < ∞, problem (4.3) has a
unique solution ϕ = T (h) and c = c(h). Moreover,
∥T (h)∥∗ ≤ C∥h∥∗ .
(4.4)
The proof of Theorem 4.2 relies heavily on the following lemma.
Lemma 4.3. Assume that qjn , j = 1, 2, · · · , k, are such that mini̸=j |qin − qjn | → ∞,
∥hn ∥∗ → 0 and that ϕn solves
∑
Lϕn = hn +
cnj Zj in R,
j
ϕn (x) → 0,
as |x| → ∞,
⟨ϕn , Zj ⟩ = 0 for j = 1, 2, · · · , k.
Then ∥ϕn ∥∗ → 0.
Proof. We will argue by contradiction. Without loss of generality we can assume
that ∥ϕn ∥∗ = 1. Our first observation is that cnj → 0. Indeed, multiplying the
equation by Zj and integrating by parts we get
∫
∑
⟨ϕn , L∗ Zj ⟩ = cnj
Zj2 +
cni ⟨Zi , Zj ⟩ + ⟨hn , Zj ⟩.
i̸=j
FRACTIONAL GIERER-MEINHARDT SYSTEM
13
Using Proposition 4.1, by rather standard calculations, it follows that cnj → 0 as
n → ∞.
Our next goal is to show
∫
W ϕn → 0 as n → ∞.
To this end consider the test function
1
Z = x · ∇W + 2W
s
and
L0 = (−∆)s + (1 − 2U )id.
We claim that
L0 Z = −2W + o(1).
2s
Indeed if we set uλ (x) = λ W (λx), then
∑
λ4s U (λx − ξi )U (λx − ξj ).
−(−∆)s uλ (x) = λ2s uλ − u2λ +
i̸=j
λ
Since Z = 1s ∂u
∂λ |λ=1 the claim follows now from Proposition 4.1 and the above.
Decompose ϕn = an W + ψn where ⟨W, ψn ⟩ = 0. Then Lψn = L0 ψn and we have
o(1) = ⟨Lϕn , Z⟩ = an ⟨LW, Z⟩ + ⟨L0 ψn , Z⟩.
But
⟨L0 ψn , Z⟩ = ⟨ψn , L0 Z⟩ = −2⟨W, ψn ⟩ + o(1) = o(1)
and
1
⟨LW, Z⟩ = ⟨W , Z⟩ + o(1) =
3s
2
∫
∫
3
x∇W + 2
2
W + o(1) = (2 − )
3s
3
∫
W 3 + o(1).
It follows that an → 0, and ⟨W, ϕn ⟩ = o(1). Going back to the equation satisfied
by ϕn , we see that
(
∑ )
(−∆)s ϕn + (1 − 2W )ϕn = o(1) W 2 +
Zj + hn ≡ gn + hn ,
j
with ∥gn ∥∗ → 0. By applying the Lemma 4.2 in [4], we get
∥ϕn ∥∗ ≤ C(∥gn ∥∗ + ∥hn ∥∗ ) = o(1)
which contradicts to the assumption ∥ϕn ∥∗ = 1. Hence we finish the proof of this
lemma.
Remark: In the statement of Lemma 4.2 in [4], it requires that µ ∈ ( 12 , 1+2s). However, we can go through the proof to get the same conclusion for the case µ = 1+2s.
Next we construct a solution to problem (4.3). To do so, we consider the following
auxiliary problem at first
{
∑k
s
(−∆)
ϕ + ϕ = g + i=1 ci Zi ,
∫
(4.5)
ϕZi = 0 for all i.
R1
Lemma 4.4. For each g with ∥g∥∗ < +∞, there exists a unique solution of problem
(4.5), ϕ =: A[g] ∈ H 2s (R). This solution satisfies
∥A[g]∥∗ ≤ C∥g∥∗ .
14
JUNCHENG WEI AND WEN YANG
For the proof of Lemma 4.4, one can see [4, Lemma 4.3].
Proof of Theorem 4.2: Let us solve problem (4.3). Let Y be the Banach space
Y := {ϕ ∈ L∞ (R1 ) | ∥ϕ∥Y := ∥ϕ∥∗ < ∞}.
(4.6)
Let A be the operator defined in Lemma 4.4. Then we have a solution to (4.3) if
we can solve
∫
(4.7)
ϕ − A[2W ϕ] + 2A[ωW 2 W ϕ] = A[g], ϕ ∈ Y.
We claim that
∫
B[ϕ] := B1 [ϕ] + B2 [ϕ] = A[2W ϕ] + A[−2ωW 2
W ϕ]
defines a compact operator in Y. At first, we show
that B2 defines a compact
∫
operator in Y. We begin to prove that C[ϕ] = ωW 2 W ϕ is a compact operator from
Y to Y. Let us assume
that ϕn is a bounded sequence in Y. By direct computations,
∫
we get I(ϕn ) = W ϕn is contained in a bounded set in R1 . Therefore, we can pick
out subsequence of I(ϕn ) converge, combined with the fact W (x) ≤ Cρ(x), we get
C is a compact operator from Y to Y. By Lemma 4.4, we have A is a continuous
operator from Y to Y. Hence, we get B2 is compact.
Then, we come to consider the operator B1 . We denote fn = A[2W ϕn ] and claim
that
|fn (x) − fn (y)|
sup
≤ C∥ϕn ∥∞ , α = min{1, 2s}.
(4.8)
|x − y|α
x̸=y
By Green representation, we have
∫
fn = A[2W ϕn ] = 2
G(x, z)W ϕn dz,
∫
Since
∥A[2W ϕn ]∥L∞ = ∥2
G(x, z)(W ϕn )(z)dz∥L∞ ≤ C∥ϕn ∥∞ ,
we can obtain
sup
x̸=y
for |x − y| ≥
1
3.
|fn (x) − fn (y)|
≤ C∥ϕn ∥∞
|x − y|α
For |x − y| < 31 , we have
∫
|fn (x) − fn (y)| ≤ C∥ϕn ∥∞
R
|G(x − z) − G(y − z)|dz.
Now, we decompose
∫
∫
|G(x − z) − G(y − z)|dz = |G(z + y − x) − G(z)|dz
R
∫R
=
|G(z + y − x) − G(z)|dz
|z|>3|y−x|
∫
+
|G(z + y − x) − G(z)|dz.
|z|≤3|y−x|
We have
∫
|z|>3|y−x|
∫
∫
1
|G(z + y − x) − G(z)|dz ≤
dt
0
|z|>3|y−x|
|∇G(z + t(y − x))|dz|y − x|,
FRACTIONAL GIERER-MEINHARDT SYSTEM
and since 3|y − x| < 1,
∫
∫
|∇G(z + t(y − x))|dz ≤ C(1 +
|z|>3|y−x|
{
≤
1>|z|>3|y−x|
|z|<3|y−x|
{
≤
dz
)
|z|2−2s
C(1 + |y − x|2s−1 ),
C(1 + log |y − x|),
On the other hand
∫
∫
|G(z + y − x) − G(z)|dz ≤ 2
|z|<4|y−x|
15
s ∈ ( 12 , 1),
s = 12 .
|G(z)|dz
C|x − y|,
C|x − y| log |x − y|,
s ∈ ( 21 , 1),
s = 12 .
Hence, we get
sup
x̸=y
|fn (x) − fn (y)|
≤ C∥ϕn ∥∞ .
|x − y|α
Therefore, (4.8) is proved.
Using the Arzela-Ascoli theorem, we can get a subsequence of fn which we
relabel the same, that converges uniformly on compact sets to a continuous function
f . Outside such a compact set, by using the property of the Green function and
standard potential analysis, we have
γn
fn = A[2W ϕn ] =
(1 + o(1)) as |x| → ∞,
|x|1+2s
∫
where γn = 2γ R W ϕn . Based on the previous subsequence
we have chosen, we can
∫
further choose a subsequence such that I(ϕn ) = R W ϕn converges, combined with
the fact ρ(x)−1 |x|−2s−1 → k as x → ∞, we can get a subsequence of fn converges
in Y outside a compact set. This implies that B1 is compact. Therefore we prove
the claim that B defines a compact operator in Y.
Finally, a priori estimate tells us that for g = 0, equation (4.3) admits only the
trivial solution. The desired result of Theorem 4.2 follows at once from Fredholm’s
alternative.
5. Basic estimates
In this section we calculate basic estimates, including error estimates. Our first
task is to analyze the solution V of the problem
(−∆)s V + ε2s V = τε
k
[∑
]2
U (|x − qi |) ,
V (x) → 0 as |x| → ∞,
i=1
where τε is given in (3.2). Denote by Z0 the solution of
(−∆)s Z0 + ε2s Z0 = U (|x|)2 ,
Z0 (x) → 0 as |x| → ∞,
and by θij (x), i ̸= j, that of
(−∆)s θij + ε2s θij = U (|x − qi |)U (|x − qj |),
θij (x) → 0 as |x| → ∞.
(5.1)
16
JUNCHENG WEI AND WEN YANG
Then we have
V (x) = τε
k
∑
Z0 (|x − qi |) + τε
i=1
∑
θij (x).
i̸=j
We will now study Z0 (|x|) in the range
x ∈ Is := [−M ε
1−2s
4s
, Mε
1−2s
4s
]
for s ∈ ( 12 , 1) and in the range
1
1
x ∈ I 21 := [−M (− log ε) 2 , M (− log ε) 2 ]
for s = 12 . Here M = 100
η .
∫
By Green representation, we have Z0 (|x|) = Gε (|x − y|)U (|y|)2 dy. We can
expand Z0 (|x|) as
)
{ ∫ ( 1−2s
1
3
1
a0 ε
+ a1 |x − y|2s−1 U (|y|)2 + O(εmin{2+ 2s −2s, 2 − 4s } ), s ∈ ( 12 , 1),
∫
∫
Z0 (|x|) =
1
− π1 log(ε|x − y|)U (|y|)2 + a2 U 2 + O(ε(− log ε) 2 ),
s = 12 .
(5.2)
∫
Let us consider the quantity Hs (x) = a1 |x − y|2s−1 U (|y|)2 for s ∈ ( 12 , 1), by
noting that U (|y|) is an even function, we can easily obtain
∫
∫
Hs (−x) = a1 | − x − y|2s−1 U (|y|)2 = a1 | − x + y|2s−1 U (|y|)2 = Hs (x).
Furthermore, we can write
Hs (|x|) = a1 |x|2s−1
∫
U (|y|)2 + fs (|x|),
where fs and its first derivative are uniformly bounded. Similarly, we can get
∫
1
H 12 (|x|) = − log |x| U (|y|)2 + f 12 (|x|),
π
where f 12 and its first derivative are uniformly bounded.
Let us now consider the function θij given in (5.1). Since θij can be represented
as
∫
θij = Gε (|x − y|)U (y − qi )U (y − qj ).
At first, we consider the case for s ∈ ( 12 , 1). For x ∈ Is , the following expansion
holds,
∫
∫
1−2s
θij (x) =a0 ε
U (y − qi )U (y − qj ) + a1 |x − y|2s−1 U (y − qi )U (y − qj )
∫
1
+ O(ε 2 ) U (y − qi )U (y − qj ).
Using Proposition 4.1 one can show that
∫
2s−1
|x − y|2s−1 U (y − qi )U (y − qj ) = O(ε 2s ),
uniformly in x ∈ Is , a similar estimate holds for the derivative of the above expression with respect to x. Let us set
∫
δs (|z|) = U (y)U (y − z)dy.
(5.3)
FRACTIONAL GIERER-MEINHARDT SYSTEM
17
Following a standard potential analysis, we have
δs (|z|) = O(1)
1
.
|z|2s+1 + 1
(5.4)
For the case s = 12 , we can expand the term θij (x) as follows
∫
1
θij (x) = −
log(ε|x − y|)U (y − qi )U (y − qj )
π
∫
1
+ a2 U (y − qi )U (y − qj ) + O(ε| log ε| 2 ).
Using Proposition 4.1 we can obtain that there is a δ > 0 and close to 0 such that
∫
1
(a2 − log |x − y|)U (y − qi )U (y − qj ) = O(
),
(log 1ε )1−δ
uniformly for x ∈ I 12 , a similar estimate holds for the derivative of the above
expression with respect to x. Let us set
∫
(5.5)
δ 12 (|z|) = U (y)U (y − z)dy.
Following a standard potential analysis, we have
1
.
1 + |z|2
δ 12 (|z|) = O(1)
(5.6)
Thus, combing the above estimates we obtain:
Lemma 5.1. For the term V , we have
(a) When s ∈ ( 12 , 1), the following estimate holds for x ∈ Is ,
V (x) = 1 + τε
k
∑
Hs (|x − qi |) + cs
i=1
When s =
1
2,
∑
δs (|qi − qj |) + O(ε2s−1 ).
i̸=j
the following estimate holds for x ∈ I 12 ,
V (x) = 1 + τε
k
∑
H 12 (|x − qi |) + c 12
i=1
∑
δ 12 (|qi − qj |) + O(
i̸=j
1
).
− log ε
Here cs , c 12 are some generic constants independent of ε
A similar estimate holds for the derivatives of
function Hs (|x|) is given in the above discussion
expansion
∫
{
a1 |x|2s−1 ∫U (|y|)2 + fs (|x|),
H(|x|) =
− π1 log |x| U (|y|)2 + f 12 (|x|),
V with respect to x. The
and, for |x| > 1, has the
s ∈ (0, 21 ),
s = 12 .
The function δs is given in (5.4) and (5.6) for the cases s ∈ ( 12 , 1) and
s = 21 respectively.
(b) When s ∈ ( 12 , 1) for x ∈ R \ Is , then we have the following lower estimate
V (x) ≥ C
1
.
1 + (ε|x|)2s+1
(5.7)
18
JUNCHENG WEI AND WEN YANG
When s = 12 , for x ∈ R \ I 12 , then we have the following lower estimate
V (x) ≥ C
1
.
1 + (ε|x|)2
(5.8)
Estimate (5.7) and (5.8) can be proven by using the potential analysis.
6. further estimates
For brevity we shall denote Ui (x) = U (x − qi ) and W =
in this section is to derive estimates for the quantity
S(W ) ≡ −(−∆)s W − W +
which can be written as
∑k
i=1
Ui . Our purpose
W2
,
V
(6.1)
W2 ∑ 2
−
Ui .
V
i=1
k
S(W ) =
Our first result is the following
Lemma 6.1. Let the number µ = 1 + 2s in the definition of ∥ · ∥∗ . For all points
qi satisfying conditions in (3.4) and all sufficiently small ε we have
{
1
Cεs− 4s ,
s ∈ ( 12 , 1),
∥S(W )∥∗ ≤
1
1−δ
C(− log ε ) , s = 12 ,
where C is some constant independent of ε and δ is any small positive number
independent of ε.
Proof. Let us assume first x ∈ Is . We write
S(W ) =
k
∑
1−V ∑ 2
Ui Uj = J1 + J2 .
Ui + 2V −1
V i=1
(6.2)
i̸=j
At first, it is easy to see in the region under consideration, we have
{
1
1 + O(εs− 4s ),
s ∈ ( 12 , 1),
V (x) =
1
1 + O( | log ε|1−δ ), s = 12 ,
where δ is any small positive number. Hence,
{
∑k
1
O(εs− 4s )( i=1 Ui2 ),
∑k
J1 =
O( | log 1ε|1−δ )( i=1 Ui2 ),
On the other hand,
s ∈ ( 12 , 1),
s = 21 .
(
)
1
1
1
+
|qi − qj |2s+1 (1 + |x − qi |)2s+1
(1 + |x − qj |)2s+1
1
= O(1)
ρ(x).
|qi − qj |2s+1
V −1 Ui Uj ≤ 2Ui Uj ≤ C
{
Hence
J2 =
1
O(εs− 4s )ρ(x),
1
O( | log
ε| )ρ(x),
s ∈ ( 12 , 1),
s = 12 .
FRACTIONAL GIERER-MEINHARDT SYSTEM
As a conclusion, for x ∈ Is , we have
{
1
O(εs− 4s )ρ(x),
|S(W )| =
O( | log 1ε|1−δ )ρ(x),
s ∈ ( 21 , 1),
s = 21 ,
19
(6.3)
for small ε.
Outside the above region, let us consider the case s ∈ ( 21 , 1) first. Assume now
x ∈ R \ Is . By using Lemma 5.1, we have
k
k
∑
(
)∑
ε2s+1 |x|2s+1 + 1
|S(W )| ≤ C ε2s+1 |x|2s+1 + 1
ρ(x)
Ui2 ≤ C
(1 + |x − qi |)1+2s
i=1
i=1
1
= O(εs− 4s )ρ(x).
(6.4)
For the case s = 21 . Assume x ∈ R \ I 21 , we get
k
k
∑
(
)∑
|S(W )| ≤ C ε2 |x|2 + 1
Ui2 ≤ C
i=1
i=1
ε2 |x|2 + 1
ρ(x)
(1 + |x − qi |)2
1
= O(−
)ρ(x)
log ε
(6.5)
Combining relations (6.3), (6.4) and (6.5), we prove the lemma.
Another quantity which will be crucial for the remaining arguments is
∫
I = S(W )Zi .
(6.6)
We shall consider i = 1 only, since the other cases are similar. Observe that
∂U (x−q1 )
1)
= − ∂U (x−q
and thus we have
∂q1
∂x
∫
−I =
(1 − V )V −1
k
∑
i=1
Ui2
∂U
(x − q1 ) +
∂x
∫
V −1
∑
Ui Uj
i̸=j
∂U
(x − q1 ) = I1 + I2 .
∂x
We will estimate I1 and I2 separately. In fact, we will find the following expansions
∂ ∑
U (qj − q1 )(1 + o(1)),
I2 = −αs
(6.7)
∂q1
j̸=1
and
{
I1 =
∑
−βs ε2s−1 ∂q∂ 1 j̸=1 |qj − q1 |2s−1 (1 + o(1)),
∑
−β 21 (− log1 ε ) ∂q∂ 1 j̸=1 log |qj − q1 |(1 + o(1)),
s ∈ ( 12 , 1),
s = 12 .
(6.8)
Here αs , βs , s ∈ [ 21 , 1) are some universal positive constants which are independent
of ε.
We will establish (6.7) at first. Using Lemma 5.1 we obtain
∫
∫ ∑
∑
∂U1
∂U1
−1
V
Ui Uj
=
(1 + o(1)).
Ui Uj
∂x
∂x
i̸=j
i̸=j
∫
1
Let us estimate Ui Uj ∂U
∂x for i ̸= j, we observe that if i, j ̸= 1, then
∫
(
)
∂U1
Ui Uj
= O (|qi − q1 ||qj − q1 |)−(1+2s) .
∂x
20
JUNCHENG WEI AND WEN YANG
On the other hand, if i = 1, j ̸= 1 we get
∫
∫
∂U1
1 ∂
U1 Uj
=−
U 2 (x − q1 )U (x − qj )
∂x
2 ∂q1
For the right hand side of the above equality, by standard potential analysis, we
can get that for a certain universal constant cs > 0 such that
∫
U 2 (x − q1 )U (x − qj ) = cs U (q1 − qj )(1 + o(1)),
with a similar estimate for its derivative. We leave the detail in the appendix.
Hence,
∫
∑
1
∂ ∑
∂U
(x − q1 ) = − cs
U (q1 − qj )(1 + o(1)),
V −1
Ui Uj
∂x
2 ∂q1
j̸=1
i̸=j
and estimates (6.7) thus follows.
Let us now consider the term I1 . Following a similar procedure, we get
∫ ∑
k
k
{∑
}
∑
I1 = −
U (x − qi )2
τε Hs (|x − qj |) + cs
δs (|qj − ql |)
i=1
j=1
j̸=l
∂
U (x − q1 )(1 + o(1)),
∂x
where Hs and δs are given in Lemma 5.1. Let us first estimate
∫
∂
gijl = U (x − qi )2 δs (|qj − ql |) U (x − q1 )dx,
∂x
×
with j ̸= l. For i = 1 this term is zero, while for i ̸= 1 we can estimate, using (5.4),
(5.6) and Lemma 5.1,
|gijl | ≤ C|qj − ql |−(2s+1) (1 + |qi − q1 |)−(2s+2) .
Then we come to consider the terms
∫
∂U
Iij = U (x − qi )2 Hs (|x − qj |)
(x − q1 ).
∂x
First we observe that the term corresponding to i = j = 1 vanishes, by symmetry.
If i and j are different, and both different from 1, then the resulting term is of lower
order, more precisely
∫
∂
Iij = U (x − qi )2 Hs (x − qj ) U (x − q1 )
∂x
∫
∂
=−
U (x − qi )2 Hs (x − qj )U (x − q1 )
∂q1
{
(
)
O(1) ∂q∂ 1 |qi − qj |2s−1 U (qi − q1 ) , s ∈ ( 12 , 1),
(
)
=
O(1) ∂q∂ 1 log |qi − qj |U (qi − q1 ) ,
s = 12 .
On the other hand, if i = 1,
∫
1 ∂
U (x)3 Hs (|x − (qj − q1 )|)dx
I1j = −
3 ∂q1
{
bs ∂q∂ 1 |q1 − qj |2s−1 (1 + o(1)), s ∈ ( 21 , 1),
=
b 12 ∂q∂ 1 log |q1 − qj |(1 + o(1)), s = 12 ,
FRACTIONAL GIERER-MEINHARDT SYSTEM
where bs are some generic constants and we used
∫
{
a0 ∫U 2 (y)dy|x|2s−1 (1 + o(1)),
Hs (x) =
− π1 U 2 (y)dy log |x|(1 + o(1)),
21
s ∈ ( 12 , 1),
s = 12 ,
provided x is sufficiently large. We put the detail in the appendix. Now, as for Ii1 ,
we get
∫
∂U
Ii1 = U (x − (qi − q1 ))2 Hs (|x|)
(x) = O(|qi − q1 |−3 ).
∂x
Combining the above estimates, we immediately get (6.8).
Hence we have found that
 ∑
(
)
∫
∂
2s−1

|qj − q1 |2s−1 (1 + o(1)),
j̸=1 ∂q1 αs U (qj − q1 ) + βs ε
(
)
S(W )Z1 =
log |qj −q1 |
∂
 ∑
1 U (qj − q1 ) + β 1
α
(1 + o(1)),
1
j̸=1 ∂q1
log
2
2
ε
s ∈ ( 21 , 1),
s = 12 .
Thus, we obtain the following result:
Lemma 6.2. For all points (q1 , q2 , · · · , qk ) satisfies (3.4). If s ∈ ( 21 , 1),
∫
∑ ∂Fs (|qj − qi |)
S(W )Zj =
(1 + o(1)),
∂qj
i̸=j
where
Fs (r) = αs U (r) + βs ε2s−1 r2s−1 .
If s = 12 ,
∫
S(W )Zj =
∑ ∂F 1 (|qj − qi |)
2
i̸=j
∂qj
(1 + o(1)),
where
F 12 (r) = α 21 U (r) + β 12
log r
.
log 1ε
7. The finite-dimensional reduction
We will carry cut the finite-dimensional reduction process sketched in the first
part of the paper. As in the previous section, we shall assume the points qi satisfy
(3.4). Recall from Section 3 that the original problem was cast in the form
(−∆)s u + u =
u2
.
T (u2 )
(7.1)
Rather than solving this directly we consider instead the problem of finding A such
that for certain constants ci one has
∑
A2
+
ci Zi
(−∆)s A + A =
(7.2)
T (A2 )
i
22
JUNCHENG WEI AND WEN YANG
and ⟨A − W, Zi ⟩ = 0 for all i. Rewriting A = W + ϕ we get that this problem is
equivalent to
∫
(−∆)s ϕ + ϕ − 2W ϕ + 2W 2 ω W ϕ
∫
∑
W2
(W + ϕ)2
W2
2
= − (−∆)s W − W +
+
−
−
2W
ϕ
+
2W
ω
Wϕ +
ci Zi
2
V
T ((W + ϕ) )
V
i
∑
= S(W ) + N (ϕ) +
ci Zi
(7.3)
and
⟨ϕ, Zi ⟩ = 0 for all i.
(7.4)
Using the operator T introduced in Theorem 4.2, we see that the problem is equivalent to finding a ϕ ∈ H so that
ϕ = T (S(W ) + N (ϕ)) ≡ Q(ϕ).
We will show that this fixed point problem has a unique solution in a region of the
form
{
1
{
∥ϕ∥∗ ≤ Cεs− 4s ,
s ∈ ( 12 , 1), }
Ds = ϕ ∈ H |
(7.5)
1
∥ϕ∥∗ ≤ C | log ε|1−δ , s = 12 ,
for any small positive constant δ, provided ε is sufficiently small. Here
H = {ϕ ∈ L∞ | ⟨ϕ, Zj ⟩ = 0, j = 1, 2, · · · , k}.
We recall that from Lemma 6.1,
{
1
∥S(W )∥∗ ≤ Cεs− 4s ,
∥S(W )∥∗ ≤ C | log 1ε|1−δ ,
s ∈ ( 12 , 1),
s = 12 .
On the other hand, N (ϕ) admits the estimate provided by the following lemma.
Lemma 7.1. Assume that ϕ ∈ Ds . Then
(
)
∥N (ϕ)∥∗ ≤ C ∥ϕ∥∗ + σ(ε) ∥ϕ∥∗
provided ε is taken sufficiently small. Here
{
1
εs− 4s , s ∈ ( 12 , 1),
σ(ε) =
1
, s = 12 ,
| log ε|1−δ
as ε → 0.
Proof. Let us assume first x ∈ R \ Is and s ∈ [ 12 , 1). We observe that using a
standard potential analysis one can show that in this range of x we have W (x) ≤
Cρ(x) and
(
)
1
1
and T (W ϕ), T (ϕ2 ) ≤ C
∥ϕ∥∗ .
T (W + ϕ)2 ≥ C
2s+1
1 + (ε|x|)
1 + (ε|x|)2s+1
Then
|N (ϕ)| ≤
[ 2W V ϕ + V ϕ2 − 2W 2 T (W ϕ) − W 2 T (ϕ2 )
∫
− 2W ϕ + 2ωW
2
V T ((W + ϕ)2 )
]
ρ(x)2
ρ(x)
≤C
+
∥ϕ∥
∥ϕ∥∗ + Cρ(x)2 ∥ϕ∥∗ .
∗
1 + (ε|x|)2s+1
1 + (ε|x|)2s+1
[
2
]
Wϕ
FRACTIONAL GIERER-MEINHARDT SYSTEM
Therefore, we get
[
]
|ρ(x)−1 N (ϕ)| ≤ C ∥ϕ∥∗ + σ(ε) ∥ϕ∥∗ .
23
(7.6)
Let us consider now the case x ∈ Is for s ∈
We decompose N (ϕ) in the
form
N (ϕ) = N1 (ϕ) + N2 (ϕ),
where
[
1
1
2T (W ϕ) ]
2T (W ϕ)
N1 (ϕ) = (W + ϕ)2
−
+
− (2W + ϕ)ϕ
2
2
T ((W + ϕ) ) V
V
V2
[ 21 , 1).
and
N2 (ϕ) = −2ϕW (1 −
[
1
) + 2W 2 ω
V
∫
Wϕ −
T (W ϕ) ] ϕ2
+
.
V2
V
We have that T ((W + ϕ)2 ) = V + 2T (W ϕ) + T (ϕ2 ). On the other hand,
{
1
1 + O(εs− 4s ),
s ∈ ( 12 , 1),
V (x) =
1
1 + O( | log ε|1−δ ), s = 12
Also,
{
T (W ϕ) =
∫
1
ω ∫ W ϕ + O(εs− 4s )∥ϕ∥∗ ,
1
ω W ϕ + O( | log ε|1−δ )∥ϕ∥∗ ,
s ∈ ( 12 , 1),
s = 12 ,
and in particular |T (W ϕ)| = O(∥ϕ∥∗ ). Likewise, T (ϕ2 ) = O(∥ϕ∥2∗ ). Combining
these facts we obtain
|N1 (ϕ)| ≤ C(W + ϕ)2 T (ϕ2 ) + C[2W ϕ + ϕ2 ]T (W ϕ)
≤ Cρ(x)∥ϕ∥2∗ .
A similar analysis yields,
{
1
Cεs− 4s (|ϕ|W + W 2 ∥ϕ∥∗ ) + C|ϕ|2 ,
|N2 (ϕ)| ≤
C
(|ϕ|W + W 2 ∥ϕ∥∗ ) + C|ϕ|2 ,
| log ε|1−δ
s ∈ ( 12 , 1),
s = 12 .
Hence,
∥N (ϕ)∥∗ ≤ C(∥ϕ∥2∗ + σ(ε)∥ϕ∥∗ )
in this region. Combining this estimate with (7.6), yields the result of the lemma.
Using the definition of the corresponding norms, splitting different ranges of x
as in the above proof, it is readily checked that the following holds: If
{
1
Cεs− 4s ,
s ∈ ( 21 , 1),
∥ϕi ∥∗ ≤
i = 1, 2,
(7.7)
1
C(− log ε )1−δ , s = 12 ,
then, given any small µ > 0, we can find ε sufficiently small such that
∥N (ϕ1 ) − N (ϕ2 )∥∗ ≤ µ∥ϕ1 − ϕ2 ∥∗ .
This implies that the operator Q is a contraction mapping in the set Ds defined
in (7.5). On the other hand, we also get from the Lemma 7.1 that Q maps Ds
into itself. By using Banach fixed point theorem, we get the existence of a unique
fixed point of Q in this domain, which depends continuously in the ∗− norm on the
points of qi . We summarize this result in the following proposition:
24
JUNCHENG WEI AND WEN YANG
Proposition 7.2. For all sufficiently small
we have the existence of a unique solution to
cs (q1 , · · · , qk ) which satisfies (7.8). Besides,
qi ’s.
In addition the following formula holds for
ε and all points qi satisfying (3.4),
(7.3), ϕs = ϕs (q1 , · · · , qk ) and cs =
(ϕs , cs ) depend continuously on the
the components of cs,j of cs :
cs,j = bs,j + es,j , j = 1, 2, · · · , 2m,
(7.8)
with
bs,j =
∑ ∂Fs (|qj − qi |)
,
∂qj
i̸=j
the error terms es,j satisfy
es,j =
{
o(1)εs+ 2 − 2s ,
(
)− 3
o(1) log 1ε 2 ,
1
1
s ∈ ( 12 , 1),
s = 12 .
Proof. We only need to prove the formula for cs,j ’s. Let us observe that the cs,j
satisfy the relations
∑
cs,j ⟨Zi , Zj ⟩ = −⟨S(W ), Zj ⟩ − ⟨N (ϕ), Zj ⟩ + ⟨ϕ, L∗ (Zj )⟩,
which define an ”almost diagonal” system, from which the cs,j ’s can be solved
uniquely. The main term in the above expansion is given by ⟨S(W ), Zj ⟩. To obtain
the estimates for these numbers, which will equal the cs,j ’s at the leading order, we
observe that
{
(2s−1)(s+1)
2s
Cε
∥ϕ∥∗ , s ∈ ( 12 , 1),
∗
|⟨ϕ, L (Zj )⟩| ≤
1
C
s = 12 ,
3 −δ ∥ϕ∥∗ ,
| log ε| 2
and
{
|⟨N (ϕ), Zj ⟩| ≤
1
Cε2s− 2s ,
C | log 1ε|2−δ ,
s ∈ ( 12 , 1),
s = 12 .
Formula (7.8) is now an immediate corollary of Lemma 6.1, Lemma 7.1, and the
expressions found for the cs,j ’s.
In the following section we will find the points qj such that all cs,j ’s vanish, and
satisfying the conditions in (3.4).
8. Solving the reduced problem
In this section, we shall look for the point (q1 , q2 , · · · , qk ) such that cs,j = 0
and thereby prove Theorem 1.1. We first establish the presence of the zero of
bs = (bs,1 , bs,2 , · · · , bs,k ) and then use the degree theory to get the existence of the
points (q1 , q2 , · · · , qk ) such that cs = 0.
We recall that
∑ ∂Fs (|qi − qj |)
,
(8.1)
bs,j =
∂qj
i̸=j
FRACTIONAL GIERER-MEINHARDT SYSTEM
where
{
αs U (r) + βs ε2s−1 r2s−1 ,
log r
α 12 U (r) + β 21 log
1 ,
Fs (r) =
ε
25
s ∈ ( 21 , 1),
s = 12 .
It is not difficult to see that finding the zero point of bs is equivalent to finding the
critical point of the following function,
)
∑(
b s (q1 , q2 , · · · , qk ) =
Ξ
αs U (qi − qj ) + βs ε2s−1 |qi − qj |2s−1
(8.2)
i̸=j
for s ∈
( 12 , 1)
and
b 1 (q1 , q2 , · · · , qk ) =
Ξ
2
∑(
α 12 U (qi − qj ) + β 12
i̸=j
)
1
1 log |qi − qj |
log ε
(8.3)
for s = 12 .
Since qi and qk+1−i are symmetry with respect to the origin for i = 1, 2, · · · , k.
b s , s ∈ [ 1 , 1) in (3.4) is reduced to finding the
So, finding the critical point of Ξ
2
critical point of Ξs introduced in (1.6) for s ∈ ( 12 , 1) in (1.8) and (1.7) for s = 12 in
(1.9) respectively.
For the functions Ξs , s ∈ [ 12 , 1), we have the following property
Lemma 8.1. The functions Ξs , s ∈ [ 12 , 1) admit an interior minimal point in the
set (1.8) and (1.9) for s ∈ ( 12 , 1) and s = 12 respectively provided η is sufficiently
small.
Proof. By Proposition 4.1, we have as |x| → ∞,
U (x) →
bs
(1 + o(1)).
|x|1+2s
For ε sufficiently small, we have that Ξs admit the following asymptotic expansion
m (
)
∑
bs αs
2s−1
2s−1
Ξs (q1 , q2 , · · · , qm ) =
(1
+
o(1))
+
β
ε
|2q
|
s
i
|2qi |1+2s
i=1
)
∑(
bs αs
2s−1
2s−1
+
(1
+
o(1))
+
β
ε
|q
−
q
|
s
i
j
|qi − qj |1+2s
i̸=j
)
∑(
bs αs
2s−1
2s−1
+
(1
+
o(1))
+
β
ε
|q
+
q
|
(8.4)
s
i
j
|qi + qj |1+2s
i̸=j
for s ∈
( 12 , 1)
and
Ξ 12 (q1 , q2 , · · · , qm ) =
m (
∑
b1 α1
2
2
i̸=j
+
|qi − qj
)
1
1 log |qi − qj |
log ε
(1 + o(1)) + β 12
2
)
1
log
|q
+
q
|
i
j
log 1ε
|2
|qi + qj |
)
1
1 log |2qi |
log ε
(1 + o(1)) + β 12
2
∑ ( b1 α1
2
2
i̸=j
for s = 12 .
(1 + o(1)) + β 21
|2qi
∑ ( b1 α1
i=1
+
2
|2
(8.5)
26
JUNCHENG WEI AND WEN YANG
For convenience, we make the following substitution,
{ 1−2s
ε 4s |di − dj |,
s ∈ ( 12 , 1),
|qi − qj | =
1 12
(log ε ) |di − dj |, s = 12 .
(8.6)
Then, (8.4)-(8.5) turns to be
m (
[∑
)
1
2s−1
(1
+
o(1))
+
γ
|2d
|
s
i
|2di |1+2s
i=1
)
∑(
1
(1 + o(1)) + γs |di − dj |2s−1
+
1+2s
|di − dj |
i̸=j
(
)]
∑
1
2s−1
+
(1
+
o(1))
+
γ
|d
+
d
|
s i
j
|di + dj |1+2s
1
Ξs (q1 , q2 , · · · , qm ) = bs αs εs− 4s
(8.7)
i̸=j
for s ∈ ( 12 , 1) and
)
1 [∑( 1
(1 + o(1)) + γ 12 log |2di |
1
2
log ε i=1 |2di |
)
∑(
1
1 log |di − dj |
+
(1
+
o(1))
+
γ
2
|di − dj |2
i̸=j
)
∑(
1
1 log |di + dj |
+
(1
+
o(1))
+
γ
2
|di + dj |2
i̸=j
1
1 ]
+ (m2 − m)γ 12 log(log )
(8.8)
2
ε
m
Ξ 21 (q1 , q2 , · · · , qm ) = b 21 α 12
for s = 12 . Here γs = βαSs .
Since the process of finding the interior global minimal point of Ξs for s ∈ ( 12 , 1)
in (1.8) and Ξs for s = 12 in (1.9) are the same, in the following we shall only give
the detail of the case s = 12 .
Before studying Ξ 12 , we first consider the following function
g(x) = x−2 + γ 12 log x for x > 0.
By analyzing the derivative of function g, we know that
√
(1 1
)
2
1
g(
) = min g(x) =
+ log 2 γ 21 − γ 21 log γ 12 .
x>0
γ 12
2 2
2
Let us come back to the function Ξ 12 . For (q1 , q2 , · · · , qm ) ∈ ∂Qs,η , we have either
there is some i such that di = η or di = η1 , or there are i, j such that |di − dj | = η.
If the former case happens, i.e., there is some i such that di = η or di = η1 . For
convenience, we write
(
Ξ̃ 21 (d1 , d2 , · · · , dm ) = b 12 α 12
1 )−1
1
1
Ξ 12 (q1 , q2 , · · · , qm ) − (m2 − m)γ 12 log log .
2
ε
log 1ε
FRACTIONAL GIERER-MEINHARDT SYSTEM
27
Then, we can get
(
)
)
1
1 (
η
Ξ̃ 12 (d1 , d2 , · · · , dm ) ≥ min{ η −2 1 + o(1) + γ 12 log 2η, η 2 1 + o(1) − γ 21 log }
4
4
2
( 2
)[( 1 1
)
](
)
1
+ 2m − m − 1
+ log 2 γ 12 − γ 21 log γ 12 1 + o(1)
2 2
2
≥ Ξ̃ 21 (1, 2, · · · , m)
(8.9)
provided η is small enough. If there are some i, j such that |di − dj | = η. Then,
(
)[( 1 1
)
](
)
1
+ log 2 γ 12 − γ 12 log γ 21 1 + o(1)
Ξ̃ 12 (d1 , d2 , · · · , dm ) ≥ 2m2 − m − 1
2
2
2
(
)
+ η −2 1 + o(1) + γ 12 log η
≥ Ξ̃ 12 (1, 2, · · · , m)
(8.10)
provided η is small enough. By (8.9) and (8.10), we get
min
q∈∂Q 1 ,η
Ξ 12 (q1 , q2 , · · · , qm ) > min Ξ 12 (q1 , q2 , · · · , qm ).
q∈Q 1 ,η
(8.11)
2
2
As a conclusion, Ξ 21 (q1 , q2 , · · · , qm ) admits a global interior minimal point in Q 12 ,η .
Before we give the proof of Theorem 1.1, we recall the following definition (see
Definition 2.4 in [19] or in [?]).
Definition 8.1. Let f : D → R be a C 1 −function, where D ⊂ Rm is an open
set. We say that x0 is stable critical point of f if ∇f (x0 ) = 0 and there exists a
neighborhood U of x0 such that
∇f (x) ̸= 0, ∀x ∈ ∂U,
∇f (x) = 0, x ∈ U
⇐⇒
f (x) = f (x0 ),
and
deg(∇f, U, 0) ̸= 0,
where deg denotes the Brouwer degree.
Remark: It is easy to see that, if x0 is a global minimum point or a global maximum point of the function f , then x0 is a stable critical point of f .
Proof of Theorem 1.1: By Lemma 8.1, we get the existence of (q1 , q2 , · · · , qk ) such
that
bs (q1 , q2 , · · · , qk ) = 0.
Furthermore, such a point is the global minimal point of the function Ξs in (1.8)
and (1.9) for s ∈ ( 12 , 1) and s = 12 respectively. We denote such point by qs =
(qs,1 , qs,2 , · · · , qs,m ). Now, we shall look for q = (q1 , q2 , · · · , qm ) in the neighborhood of qs to make cs = 0. As we mentioned in the previous remark, such a global
minimal point is a stable critical point of Ξs . Using the property of the stable
b s and Ξs , we can find an open
critical point and relation between the function Ξ
neighborhood Oqs of qs in Qs,η such that the following holds
(
)
b s , Oq , 0 ̸= 0 and ∇q Ξ
b s ̸= 0 on ∂Oq ,
deg ∇qs Ξ
s
s
s
28
JUNCHENG WEI AND WEN YANG
which implies
(
)
deg bs , Oqs , 0 ̸= 0 and bs ̸= 0 on ∂Oqs .
According to the definition of Ξs , we can get
{
1
1
ds εs+ 2 − 2s , s ∈ ( 12 , 1),
1
|bs | ≥
s = 12 ,
d 21
3 ,
1
(log
ε)
on ∂Oqs ,
2
where ds , s ∈ [ 12 , 1) are strictly positive constants.
Next, Let us introduce the following homotopy,
H(t, s, q) = bs + t(cs − bs ).
We find that
H(1, s, q) = cs and H(0, s, q) = bs .
It is known in Proposition 7.2
{
1
1
o(1)εs+ 2 − 2s ,
|cs − bs | =
(
)
−3
o(1) log 1ε 2 ,
s ∈ ( 12 , 1),
s = 12 ,
in Oqs .
As a result, we get H(t, s, q) ̸= 0 on ∂Oqs . Therefore,
(
)
(
)
deg H(1, s, q), Oqs , 0 = deg H(0, s, q), Oqs , 0 ,
which implies
(
)
(
)
deg cs , Oqs , 0 = deg bs , Oqs , 0 .
We already
( know that
) the right hand side of the above equality is non-zero, therefore, deg cs , Oqs , 0 ̸= 0. As a result, we can find q in Oqs such that cs = 0. Hence,
we finish the proof of Theorem 1.1.
The proof of even number bumps case is thus concluded. Let as assume that
k = 2m + 1 and briefly sketch the way to proceed in this situation. In this case we
introduce the set
{
(q1 , q2 , · · · , qk ) ∈ Λos = (q1 , q2 , · · · , qk ) | qk = 0, qi = −qk−i , q1 > q2 > · · · > qk−1 ,
}
(q1 , q2 , · · · , qm ) ∈ Qs,η .
Now we consider the first approximation
W (x) =
k
∑
U (x − qi ), (q1 , q2 , · · · , qk ) ∈ Λos .
i=1
In this case, because of the even symmetry, the ”bad directions” corresponding to
small eigenvalues are only those Zi with 1 ≤ i ≤ k − 1. With this observation made,
the rest of the scheme of proof goes almost the same way.
FRACTIONAL GIERER-MEINHARDT SYSTEM
29
9. Appendix
In this section, we list the estimates used in previous sections and give a proof
in the following lemma.
Lemma 9.1.
∫
U 2 (|y|)U (|x − y|)dy = cs U (|x|)(1 + o(1)) as |x| → ∞,
(9.1)
U 2 (|y|)|x − y|2s−1 dy = cs |x|2s−1 (1 + o(1)) as |x| → ∞,
(9.2)
U 2 (|y|) log |x − y|dy = cs log |x|(1 + o(1)) as |x| → ∞.
(9.3)
∫
and
∫
∫
Here cs depends on the integral of
U 2.
Proof. Since the proof of (9.1)-(9.3) are the same, we only give the proof of the first
one. We divide the whole space into two parts,
2
2
R = I1 ∪ I2 := {y : |y| ≤ |x| 3 } ∪ {y : |y| > |x| 3 }.
Then, for any y ∈ I1 , we have
1
1 (
1
1
y )
=
=
1 + O( ) ,
|x − y|2s+1 |x|2s+1 |1 − xy |2s+1
|x|2s+1
x
and
∫
∫
2
U (y) dy =
I1
∫
2
U (y) dy + O(
R1
I2
− 8s+2
3
=c0 + O(|x|
1
dy)
|y|4s+2
),
∫
s
where we used U (y) → |y|b1+2s
as |y| → ∞ and c0 = R1 U (y)2 dy. Thus,
∫
∫
∫
(
)
U 2 (|y|)U (|x − y|)dy =U (x)
U (y)2 +
U (x − y) − U (x) U (y)2
I1
I1
=c0 U (|x|) + o(1)(|x|
I1
−2s−1
).
(9.4)
For y ∈ I2 , we have
U (y)2 U (x − y) ≤ C
Hence,
∫
U (y)2 U (x − y) = O(|x|−
1
|y|4s+2
8s+4
3
U (x − y).
) = O(|x|−2s−1−
2s+1
3
).
(9.5)
I2
Combining (9.4) and (9.5), we get (9.1).
Acknowledgments. J. Wei is partially supported by NSERC of Canada.
30
JUNCHENG WEI AND WEN YANG
References
[1] C. J. Amick, J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono
equationa nonlinear Neumann problem in the plane, Acta Math. 167 (1991) 107126.
[2] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm.
Partial Differential Equations 32 (7-9) (2007) 1245-1260.
[3] X. Cabré, J. G. Tan, Positive solutions of nonlinear problems involving the square root of
the Laplacian, Adv. Math. 224(2010), no.5, 2052-2093.
[4] J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear
Schrödinger equation, J. Differential Equations 256 (2014), no. 2, 858-892.
[5] A. Doelman, R. A. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion
equations, Indiana Univ. Math. J. 50 (5) (2001) 443-507.
[6] A. Doelman, T. J. Kaper and H. van der Ploeg, Spatially periodic and aperiodic multi-pulse
patterns in the one-dimensional Gierer-Meinhardt equation, Methods Appl. Anal. 8 (2001),
387414.
[7] M. del Pino, M. Kowalczyk and X. F. Chen, The Gierer-Meinhardt system: the breaking
of homoclinics and multi-bump ground states. Commun. Contemp. Math. 3 (2001), no.3,
419-439.
[8] M. Del Pino, M. Kowalczyk and J. Wei, Multi-bump ground states of the Gierer-Meinhardt
system in R2 . Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 1, 53-85.
[9] R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians
in R. Acta Math. 210 (2013), no. 2, 261-318.
[10] R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional
Laplacian, preprint arXiv: 1302.2652v1.
[11] P. Felmer, A. Quaas and J. G. Tan, Positive solutions of the nonlinear Schrdinger equation
with the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), no. 6, 1237-1262.
[12] A. Gierer, H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin) 12
(1972) 30-39.
[13] Golovin, A. A.; Matkowsky, B. J.; Volpert, V. A. Turing pattern formation in the Brusselator
model with superdiffusion. SIAM J. Appl. Math. 69 (2008), no. 1, 251272. 35K57
[14] B. I. Henry, T. A. M. Langlands and S.L. Wearne, Turing pattern formation in fractional
activator-inhibitor systems, Physical Review E vol.72, no.2, Article ID 026101, 14 pages,
2005.
[15] D. Iron, M. Ward and J. Wei, The stability of spike solutions to the one-dimensional GiererMeinhardt model, Phys. D 150 (2001), 2562.
[16] V. Méndez D. Campos and J. Fort, Dynamical features of reaction-diffusion fronts in fractals,
Physical Review E vol.69, no.2, Article ID 016613, 7 pages, 2004.
[17] H. Meinhardt, The Algorithmic Beauty of Sea Shells, 2nd Edition, Springer, Berlin, 1998.
[18] H. Meinhardt, Models of Biological Pattern Formation, Academic Press, London, 1982.
[19] M. Musso, A. Pistoia, Multispike solutions for a nonlinear elliptic problem involving the
critical Sobolev exponent. Indiana Univ. Math. J. 51(3), (2002) 541-579.
[20] Kolokolnikov, Theodore; Wei, Juncheng; Yang, Wen On large ring solutions for GiererMeinhardt system in R3 . J. Differential Equations 255 (2013), no. 7, 14081436.
[21] T. Kolokolonikov, J. Wei, Positive clustered layered solutions for the GiererMeinhardt system,
J. Differential Equations 245 (4) (2008) 964993.
[22] T. Kolokolonikov, X. Ren, Smoke-ring solutions of GiererMeinhardt System in R3 , SIAM J.
Appl. Dyn. Syst. 10 (1) (2011) 251277.
[23] Y. Nec, Spike-like solutions to one dimensional Gierer-Meinhardt model with Levy flights,
Stud. Appl. Math. 129(2012), no.3, 272-299.
[24] W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices of Amer.
Math. Soc. 45 (1998) 9-18.
[25] W.-M. Ni, J. Wei, On positive solutions concentrating on spheres for the GiererMeinhardt
system, J. Differential Equations 221 (1) (2006) 158189.
[26] X. Ros-Oton, J. Serra, The Pohozaev identity for the fractional Laplacian. Arch. Ration.
Mech. Anal. 213 (2014), no. 2, 587-628.
[27] A. M. Turing, The chemical basis of a morphogenesis, Philos. Trans. Roy. Soc. London Ser.
B 237 (1952) 37-72.
FRACTIONAL GIERER-MEINHARDT SYSTEM
31
[28] M. J. Ward and J. Wei, Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt
model: Equilibria and stability, European J. Appl. Math. 13 (2002), 283320.
[29] J. Wei, Uniqueness and eigenvalue estimates of boundary spike solutions, Proc. Roy. Soc.
Edinburgh Sect. A 131 (2001) 1457-1480.
[30] J. Wei, On single interior spike solutions of Gierer-Meinhardt system: uniqueness and spectrum estimates, European J. Appl. Math. 10 (1999) 353-378.
[31] J. Wei, Point-condensations generated by Gierer-Meinhardt system: a brief survey, in: Y.
Morita, H. Ninomiya, E. Yanagida, S. Yotsutani (Eds.), New Trends in Nonlinear Partial
Differential Equations, 2000, pp. 46-59.
[32] J. Wei, M. Winter, On the two-dimensional Gierer-Meinhardt system with strong coupling,
SIAM J. Math. Anal. 30 (1999) 1241-1263.
[33] J. Wei, M. Winter, On multiple spike solutions for the two-dimensional Gierer-Meinhardt
system: the strong coupling case, J. Differential Equations 178 (2002) 478-518.
[34] J. Wei, M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, J. Nonlinear Science 6 (2001) 415-458.
[35] J. Wei, Existence and stability of spikes for the Gierer-Meinhardt system, Handbook of
Differential Equations-stationary partial differential equations, Volume 5 (M. Chipot Ed.),
Elsevier, pp. 489-581.
[36] J. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems
Applied Mathematical Sciences Series, Vol. 189, Springer 2014, ISBN: 978-4471-5525-6.
Juncheng Wei, Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
E-mail address: jcwei@math.ubc.ca
Wen Yang, Department of Mathematics, University of British Columbia, Vancouver,
BC V6T 1Z2, Canada
E-mail address: wyang@math.ubc.ca